Finite Group Databasebeta

← #10594220

Subgroups — #10594220

Order 576.

519 conjugacy classes of subgroups.

The subgroup lattice has 499 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#64095C211yesmaximal, characteristic, complemented
2C3×#884C211yesmaximal, characteristic, complemented
3C3×#883C211yesmaximal, characteristic, complemented
4#63980C211yesmaximal, characteristic, complemented
5#63979C211yesmaximal, characteristic, complemented
6#67⋊Dic3C211yesmaximal, characteristic, complemented
7#63958C211yesmaximal, characteristic, complemented
8#63908C211yesmaximal, characteristic, complemented
9#63906C211yesmaximal, characteristic, complemented
10#63904C211yesmaximal, characteristic, complemented
11#63891C211yesmaximal, characteristic, complemented
12#63886C211yesmaximal, characteristic, complemented
13#63854C211yesmaximal, characteristic, complemented
14#63845C211yesmaximal, characteristic, complemented
15#63841C211yesmaximal, characteristic, complemented
16#5663—31nomaximal
17#5619—31nomaximal
18C2×#638C2211yescharacteristic
19C22×C3⋊Dic3C2211yescharacteristic
20C2×C3×#67C2211yescharacteristic
21C2×C3×#67C2211yescharacteristic
22C2×C3×#67C2211yescharacteristic
23C2×C3×D12C2211yescharacteristic
24C2×C3×C4×D3C2211yescharacteristic
25C2×C3×C4×D3C2211yescharacteristic
26C2×#626C2211yescharacteristic
27C2×#626C2211yescharacteristic
28C2×#626C2211yescharacteristic
29C2×#625C2211yescharacteristic
30C2×#625C2211yescharacteristic
31C2×#624C2211yescharacteristic
32C2×D3×Dic3C2211yescharacteristic
33C32×C22⋊C4C2211yescharacteristic
34#3752C2211yescharacteristic
35#3752C2211yescharacteristic
36C4⋊(C3⋊Dic3)C2211yescharacteristic
37C4⋊(C3⋊Dic3)C2211yescharacteristic
38C3×#217C2211yescharacteristic
39C3×(C2×D3)⋊C4C2211yescharacteristic
40C3×(C2×D3)⋊C4C2211yescharacteristic
41C3×C4⋊Dic3C2211yescharacteristic
42C3×Dic3⋊C4C2211yescharacteristic
43C3×Dic3⋊C4C2211yescharacteristic
44#3724C2211yescharacteristic
45Dic3⋊Dic3C2211yescharacteristic
46Dic3⋊Dic3C2211yescharacteristic
47(C2×D3)⋊Dic3C2211yescharacteristic
48(C2×D3)⋊Dic3C2211yescharacteristic
49(C2×D3)⋊Dic3C2211yescharacteristic
50(C2×D3)⋊Dic3C2211yescharacteristic
51Dic32C2211yescharacteristic
52Dic32C2211yescharacteristic
53C3×#123—31no—
54C3×#121—31no—
55#940—31no—
56#940—31no—
57#939—31no—
58#939—31no—
59#939—31no—
60#938—31no—
61#938—31no—
62D4×Dic3—31no—
63#930—31no—
64#930—31no—
65C4×#67—31no—
66#926—31no—
67#895—31no—
68#894—31no—
69#893—31no—
70#892—31no—
71#890—31no—
72#889—31no—
73#886—31no—
74#886—31no—
75#886—31no—
76#886—61no—
77#884—31no—
78#884—31no—
79#884D311yescharacteristic, complemented
80#883—31no—
81#883—31no—
82#883D311yescharacteristic, complemented
83#67⋊C4—31no—
84C23×C32C2311yescharacteristic
85C22×C3⋊D3C2311yescharacteristic
86C22×C3×D3C2311yescharacteristic
87C22×C3×D3C2311yescharacteristic
88C22×C3×D3C2311yescharacteristic
89C2×C32×C4C2311yescharacteristic
90C2×C32×C4C2311yescharacteristic
91#638—41no—
92C2×C3⋊Dic3—41no—
93C2×C3⋊Dic3C2311yescharacteristic
94C2×C3⋊Dic3C2311yescharacteristic
95C2×C3⋊Dic3C2311yescharacteristic
96C3×#67—22no—
97C3×#67—41no—
98C3×#67—41no—
99C2×C3×Dic3C2311yescharacteristic
100C2×C3×Dic3C2311yescharacteristic
101C2×C3×Dic3C2311yescharacteristic
102C2×C3×Dic3C2311yescharacteristic
103C2×C3×Dic3C2311yescharacteristic
104C3×D12—41no—
105C3×C4×D3—22no—
106C3×C4×D3—41no—
107#626—22no—
108#626—41no—
109#626—41no—
110#625—41no—
111#625—41no—
112#624—41no—
113D3×Dic3—41no—
114#540—91noSylow
115C2×C3×D4—31no—
116C2×C3×D4—31no—
117C2×C3×D4—31no—
118C2×C3×D4—31no—
119C22×C3×C4—31no—
120C22×C3×C4—31no—
121C2×#67—31no—
122C2×#67—31no—
123C2×#67—31no—
124C2×#67—31no—
125C2×#67—31no—
126C2×#67—31no—
127C2×#67—31no—
128C2×#67—31no—
129C2×#67—31no—
130C2×#67—61no—
131C2×#67C2×D311yescharacteristic
132C2×#67C2×D311yescharacteristic
133C2×#67C2×D311yescharacteristic
134C22×Dic3—31no—
135C22×Dic3—31no—
136C22×Dic3—31no—
137C22×Dic3—61no—
138C2×D12—31no—
139C2×D12—31no—
140C2×D12—31no—
141C2×D12C2×D311yescharacteristic
142C2×C4×D3—31no—
143C2×C4×D3—31no—
144C2×C4×D3—31no—
145C2×C4×D3C2×D311yescharacteristic
146C2×C4×D3C2×D311yescharacteristic
147C3×C4⋊C4—31no—
148C3×C4⋊C4—31no—
149C3×C4⋊C4—31no—
150C3×C22⋊C4—21no—
151C3×C22⋊C4—31no—
152C3×C22⋊C4—31no—
153C3×C22⋊C4—31no—
154C3×C22⋊C4C2×D311yescharacteristic
155C3×C22⋊C4C2×D311yescharacteristic
156#217—31no—
157#217—31no—
158#217—31no—
159#217—31no—
160#217C2×D311yescharacteristic
161(C2×D3)⋊C4—31no—
162(C2×D3)⋊C4—31no—
163(C2×D3)⋊C4—31no—
164(C2×D3)⋊C4—31no—
165(C2×D3)⋊C4—31no—
166(C2×D3)⋊C4—31no—
167(C2×D3)⋊C4—31no—
168(C2×D3)⋊C4—31no—
169(C2×D3)⋊C4—61no—
170(C2×D3)⋊C4—61no—
171(C2×D3)⋊C4C2×D311yescharacteristic
172(C2×D3)⋊C4C2×D311yescharacteristic
173C4⋊Dic3—31no—
174C4⋊Dic3—31no—
175C4⋊Dic3—31no—
176C4⋊Dic3—31no—
177C4⋊Dic3—31no—
178C4⋊Dic3—31no—
179C4⋊Dic3—61no—
180C4⋊Dic3—61no—
181C4⋊Dic3C2×D311yescharacteristic
182Dic3⋊C4—31no—
183Dic3⋊C4—31no—
184Dic3⋊C4—31no—
185Dic3⋊C4—31no—
186Dic3⋊C4C2×D311yescharacteristic
187Dic3⋊C4C2×D311yescharacteristic
188C4×Dic3—31no—
189C4×Dic3—31no—
190C4×Dic3—31no—
191C4×Dic3—31no—
192C22×C32—21no—
193C22×C32—21no—
194C22×C32—21no—
195C22×C32C2411yescharacteristic
196C2×C3⋊D3—21no—
197C2×C3⋊D3—21no—
198C2×C3⋊D3—21no—
199C2×C3×D3—21no—
200C2×C3×D3—21no—
201C2×C3×D3—21no—
202C2×C3×D3—21no—
203C2×C3×D3—21no—
204C2×C3×D3—21no—
205C2×C3×D3—21no—
206C2×C3×D3—21no—
207C2×C3×D3—21no—
208C32×C4—21no—
209C32×C4—21no—
210C3⋊Dic3—21no—
211C3⋊Dic3—21no—
212C3⋊Dic3—21no—
213C3×Dic3—21no—
214C3×Dic3—21no—
215C3×Dic3—21no—
216C3×Dic3—21no—
217C3×Dic3#4112yes—
218#127—91no—
219#125—91no—
220#123—31no—
221#123—91no—
222#123—91no—
223#123—91no—
224#121—31no—
225#121—91no—
226#121—91no—
227#121—91no—
228#121—91no—
229#121—91no—
230C4×D4—91no—
231C4×D4—91no—
232#117—91no—
233C23×C3—21no—
234C23×C3—31no—
235C23×C3—31no—
236C23×C3—31no—
237C23×C3C22×D311yescharacteristic
238C23×C3C22×D311yescharacteristic
239C22×D3—31no—
240C22×D3—31no—
241C22×D3—61no—
242C22×D3C22×D311yescharacteristic
243C22×D3C22×D311yescharacteristic
244C22×D3C22×D311yescharacteristic
245C3×D4—62no—
246C3×D4—121no—
247C3×D4—121no—
248C3×D4—121no—
249C2×C3×C4—21no—
250C2×C3×C4—21no—
251C2×C3×C4—31no—
252C2×C3×C4—31no—
253C2×C3×C4—31no—
254C2×C3×C4—31no—
255C2×C3×C4—31no—
256C2×C3×C4—62no—
257C2×C3×C4—121no—
258C2×C3×C4C22×D311yescharacteristic
259C2×C3×C4C22×D311yescharacteristic
260C2×C3×C4C22×D311yescharacteristic
261C2×C3×C4C22×D311yescharacteristic
262#67—22no—
263#67—41no—
264#67—41no—
265#67—62no—
266#67—121no—
267#67—121no—
268#67—121no—
269#67—121no—
270#67—121no—
271#67—121no—
272#67—121no—
273#67—121no—
274#67—122no—
275C2×Dic3—31no—
276C2×Dic3—31no—
277C2×Dic3—31no—
278C2×Dic3—31no—
279C2×Dic3—31no—
280C2×Dic3—31no—
281C2×Dic3—61no—
282C2×Dic3—61no—
283C2×Dic3—61no—
284C2×Dic3—121no—
285C2×Dic3—121no—
286C2×Dic3—121no—
287C2×Dic3—122no—
288C2×Dic3C22×D311yescharacteristic
289C2×Dic3C22×D311yescharacteristic
290C2×Dic3C22×D311yescharacteristic
291C2×Dic3C22×D311yescharacteristic
292C2×Dic3C22×D311yescharacteristic
293D12—41no—
294D12—62no—
295D12—121no—
296D12—121no—
297C4×D3—22no—
298C4×D3—41no—
299C4×D3—121no—
300C4×D3—121no—
301C4×D3—121no—
302C2×C32—41no—
303C2×C32C2×#4111yescharacteristic
304C2×C32H(5,𝔽2)11yescharacteristic
305C2×C32H(5,𝔽2)11yescharacteristic
306C3⋊D3—41no—
307C3×D3—41no—
308C3×D3—41no—
309C3×D3—41no—
310C2×D4—31no—
311C2×D4—31no—
312C2×D4—31no—
313C2×D4—31no—
314C2×D4—91no—
315C2×D4—91no—
316C2×D4—91no—
317C2×D4—91no—
318C2×D4—91no—
319C2×D4—91no—
320C22×C4—31no—
321C22×C4—31no—
322C22×C4—91no—
323C22×C4—91no—
324C22×C4—91no—
325C4⋊C4—31no—
326C4⋊C4—31no—
327C4⋊C4—31no—
328C4⋊C4—91no—
329C4⋊C4—91no—
330C4⋊C4—91no—
331C4⋊C4—91no—
332C4⋊C4—91no—
333C22⋊C4—31no—
334C22⋊C4—31no—
335C22⋊C4—31no—
336C22⋊C4—91no—
337C22⋊C4—91no—
338C22⋊C4—91no—
339C22⋊C4—91no—
340C22⋊C4—91no—
341C22⋊C4—91no—
342C22⋊C4D3211yescharacteristic
343C42—91no—
344C42—91no—
345C22×C3—22no—
346C22×C3—21no—
347C22×C3—21no—
348C22×C3—21no—
349C22×C3—21no—
350C22×C3—21no—
351C22×C3—21no—
352C22×C3—21no—
353C22×C3—41no—
354C22×C3—41no—
355C22×C3—61no—
356C22×C3—61no—
357C22×C3—61no—
358C22×C3—61no—
359C22×C3—61no—
360C22×C3—61no—
361C22×C3—61no—
362C22×C3—61no—
363C22×C3—61no—
364C22×C3C23×D311yescharacteristic
365C22×C3C23×D311yescharacteristic
366C2×D3—21no—
367C2×D3—21no—
368C2×D3—21no—
369C2×D3—21no—
370C2×D3—21no—
371C2×D3—21no—
372C2×D3—21no—
373C2×D3—21no—
374C2×D3—21no—
375C2×D3—61no—
376C2×D3—61no—
377C2×D3—61no—
378C2×D3—61no—
379C2×D3—61no—
380C2×D3—61no—
381C2×D3—121no—
382C2×D3—121no—
383C2×D3—121no—
384C3×C4—21no—
385C3×C4—21no—
386C3×C4—21no—
387C3×C4—21no—
388C3×C4—32no—
389C3×C4—41no—
390C3×C4—41no—
391C3×C4—61no—
392C3×C4—61no—
393C3×C4—61no—
394C3×C4—61no—
395Dic3—21no—
396Dic3—21no—
397Dic3—21no—
398Dic3—21no—
399Dic3—61no—
400Dic3—61no—
401Dic3—61no—
402Dic3—61no—
403Dic3—61no—
404Dic3—61no—
405Dic3—121no—
406Dic3—121no—
407Dic3—121no—
408Dic3#23712yes—
409C32#54011yescharacteristic, complemented, Sylow
410C23—31no—
411C23—31no—
412C23—31no—
413C23—91no—
414C23C2×D3211yescharacteristic
415D4—62no—
416D4—121no—
417D4—121no—
418D4—121no—
419D4—182no—
420D4—361no—
421D4—361no—
422D4—361no—
423D4—361no—
424D4—361no—
425C2×C4—31no—
426C2×C4—31no—
427C2×C4—31no—
428C2×C4—31no—
429C2×C4—31no—
430C2×C4—62no—
431C2×C4—91no—
432C2×C4—91no—
433C2×C4—91no—
434C2×C4—121no—
435C2×C4—361no—
436C2×C4—361no—
437C2×C4—361no—
438C2×C4C2×D3211yescharacteristic
439C2×C4C2×D3211yescharacteristic
440C2×C3—21no—
441C2×C3—21no—
442C2×C3—21no—
443C2×C3—42no—
444C2×C3—41no—
445C2×C3—41no—
446C2×C3—121no—
447C2×C3—121no—
448C2×C3—121no—
449C2×C3C2×#23711yescharacteristic
450C2×C3#100411yescharacteristic
451C2×C3#100411yescharacteristic
452C2×C3D3×#4111yescharacteristic
453C2×C3#100911yescharacteristic
454C2×C3#100911yescharacteristic
455D3—41no—
456D3—41no—
457D3—41no—
458D3—121no—
459D3—121no—
460D3—241no—
461C22—21no—
462C22—21no—
463C22—21no—
464C22—61no—
465C22—61no—
466C22—61no—
467C22—61no—
468C22—61no—
469C22—61no—
470C22—61no—
471C22—61no—
472C22—61no—
473C22—181no—
474C22—181no—
475C22—181no—
476C22C22×D3211yescharacteristic
477C4—21no—
478C4—21no—
479C4—32no—
480C4—61no—
481C4—61no—
482C4—61no—
483C4—61no—
484C4—181no—
485C4—181no—
486C4—181no—
487C3—21no—
488C3#561911yescharacteristic, complemented
489C3#566311yescharacteristic, complemented
490C2—41no—
491C2—121no—
492C2—121no—
493C2—121no—
494C2—361no—
495C2#6430811yescharacteristic
496C2D3×#23711yescharacteristic
497C2#6431311yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC32×C22⋊C4C22
hypercenterC22⋊C4D32
hypocenterC32#540
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #64095C2semidirect
2. C3×#884C2semidirect
3. C3×#883C2semidirect
4. #63980C2semidirect
5. #63979C2semidirect
6. #67⋊Dic3C2semidirect
7. #63958C2semidirect
8. #63908C2semidirect
9. #63906C2semidirect
10. #63904C2semidirect
11. #63891C2semidirect
12. #63886C2semidirect
13. #63854C2semidirect
14. #63845C2semidirect
15. #63841C2semidirect
79. #884D3semidirect
82. #883D3semidirect
409. C32#540semidirect
488. C3#5619semidirect
489. C3#5663semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#640952
C3×#8842
C3×#8832
#639802
#639792
#67⋊Dic32
#639582
#639082
#639062
#639042
#638912
#638862
#638542
#638452
#638412
#56633
#56193